Both sides are focused on getting a deal by October that can be ratified by the UK Parliament and the EU member states before Britain officially leaves on 29 March next year.

But they are also stepping up preparations for the consequences of talks breaking down without a deal.

Lord Price, who quit the government in September, said: “What you will see is, rather than a pinch on supply – although that is highly likely – a pretty significant increase in the cost of fruit and veg, the cost of meat and the cost of dairy products.”

He said the UK only produces about 25% of the fruit and vegetables it consumes and, while the winter season for imports from places like South America and New Zealand could be extended slightly to cover the UK’s EU departure, supermarkets would have to find new supply routes.

France’s European affairs minister Nathalie Loiseau told Today a “no deal” Brexit would mean “traffic jams in Calais and in each and every European port welcoming goods and people coming from the United Kingdom”.

She added: “We would all suffer. The worst would be for the United Kingdom.”

Some UK ministers have dismissed talk of food shortages in the shops in the event of there being no deal.

“I am not aware of any plans for stockpiling food. It seems to me to be a fairly ridiculous scare story,” Brexit Minister Lord Callanan told the House of Lords last week.

“There are many countries outside of the European Union that manage to feed their citizens perfectly satisfactorily without the benefit of EU processes.”

Brexit Secretary Dominic Raab told MPs on Tuesday the government would take steps to ensure an “adequate food supply”.

In an interview with Channel 5’s 5 News on Wednesday, Prime Minister Theresa May did not deny that stockpiling was happening, saying the government is being “responsible and sensible” while still trying to get a good deal with the EU.

Mrs May has taken personal control of Brexit negotiations, which are being run by a unit in the Cabinet Office reporting directly to her.

Mr Raab is holding face-to-face talks with Mr Barnier at the end of the latest round of Brexit negotiations, along with senior civil servant Olly Robbins.